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<p><dfn class="terminology">Example 1</dfn>Is <span class="process-math">\(x=0\)</span> an ordinary point or singular point of</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
x y^{\prime \prime}+(\sin x) ~y^{\prime}+x^3 y=0?
\end{equation*}
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<p class="continuation"><dfn class="terminology">Solution:</dfn></p>
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\begin{equation*}
\begin{aligned}
\frac{Q(x)}{P(x)}&amp;=\frac{\sin x}{x}=\frac{1}{x} \left[x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots \right]\\
&amp;=1-\frac{x^2}{3!}+\frac{x^4}{5!}+\cdots,\quad |x|&lt;\infty,\\
\frac{R(x)}{P(x)}&amp;=x^2.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Since both <span class="process-math">\(Q(x)/P(x)\)</span> and <span class="process-math">\(R(x)/P(x)\)</span> are analytic at <span class="process-math">\(x=0\text{,}\)</span> <span class="process-math">\(x=0\)</span> is an ordinary point.</p>
<span class="incontext"><a href="sec5_2.html#p-208" class="internal">in-context</a></span>
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